Question

Solve : $\frac{2}{x}+\frac{3}{y}=13$, $\frac{5}{x}-\frac{4}{y}=-2$

Answer 1

Solving the equations $\frac{2}{x}+\frac{3}{y}=13$ and $\frac{5}{x}-\frac{4}{y}=-2$

The given equations are as follows

$\frac{2}{x}+\frac{3}{y}=13...\left(1\right)$

$\frac{5}{x}-\frac{4}{y}=-2...\left(2\right)$

Substitute$\frac{1}{x}=a$ and $\frac{1}{y}=b$, we get the following pair of linear equations:

$$\begin{gathered} 2a+3b=13.........\left(3\right) \\ 5a–4b=-2..........\left(4\right) \end{gathered}$$

Step 1: Making same coefficient of $b$ in equation $\left(3\right)$ and $\left(4\right)$

Coefficients of $b$ are $3$and $4$ and their LCM is $12$.

Using the elimination method .

Equation $\left(3\right)\times 4$

$8a+12b=52.............\left(5\right)$

Equation $\left(4\right)\times 3$

$15a–12b=-6............\left(6\right)$

Step 2: Finding the value of $a$

Add equations $5$ and $6$, we get

$$\begin{gathered} 23a=46 \\ a=2 \end{gathered}$$

Step 3: Finding the value of $b$

Substitute the value of $a$ in equation $\left(3\right)$, we get

$$\begin{gathered} 2\times 2+3b=13 \\ \Rightarrow 4+3b=13 \\ \Rightarrow 3b=13–4 \\ \Rightarrow 3b=9 \\ \Rightarrow b=\frac{9}{3} \\ \Rightarrow b=3 \end{gathered}$$

Step 4: Finding the value of $x$ and $y$

We know that

$\frac{1}{x}=a=2$

So, $x=\frac{1}{2}$

Similarly, $\frac{1}{y}=b=3$

So, $y=\frac{1}{3}$

Hence , $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is the correct answer.